On sequences of Moebius transformations

DE PREE, J. D., and W. J. THRON

Math. Zeitschr. 80, ~84-- 193 (1962)

On Sequences of Moebius Transformations* By

J. D. DE PREE and W. J. TItROU

1. Introduction Let

P z + Q P S - - Q R ~ O T(z) - - R z + S '

be a Moebius transformation. We will consider Sequences {T~} of Moebius transformations. The following theorem, which completely classifies the convergence behavior of such sequences, has been proved by PIRANIAN and TaRON [2] :

THEOREM t.1. For sequences {T~) of Moebius trans]ormations, the convergence behavior can be divided into the [ollowing cases, each o] which can o c c u r ."

(a) the limit ]unction is a Moebius trans]ormation, and the set o/convergence is the entire #lane,"

(b) lim T . ( z ) = I ~, a constant, ]or all z:4=zo;

(c) the limi t /unction takes on precisely two distinct values and the set o/ convergence consists o /only two points,"

(d) with the exception o] case (b), the limit ]unction is a constant and the set o] convergence is any "permissible set".

(e) the set o] convergence is the null set.

W e are particularly interested in sequences {T.} that are generated by functional composition from s e q u e n c e s {tn} , of Moebius transformations in the following way:

T l - ~ t l , T . -~T . :~Ot . , (n----e, 3 . . . . )

where the symbol "o ~' denotes functional composition. I t is clear that the T~ are also Moebius transformations. If we wri~e-

t~,(z) - ~ + ~" a ~ d ~ . - b , - % 4 = o , Cn z + d n ' . .

R~ z + S~ '

then the parameters P~,. Q., R~, S~ are determined only up to a common multiplicative constant, say M~. By suitable choice of M.,, we can obtain

�9 This research was supported by t h e United States Air Force under contract No. AF 49{638)-t00.

On Sequences of Moebius Transformations 185

P., Q~, R~, S. from the coefficients a . , b., c., d~ of t~ by the following r e - cursion formulas:

Pl -m- a l , QI : bl l R I : Cl, S : : d 1 ,

(t .1) R~=a .R~2:+cnS~_: , S , ,=b~R~_I+d.S~_ I , (n =2 , 3 . . . . ).

Immediate consequences of (t.t) are

and (s~_~ P~ - O.-x R.) ~ + (Sn_~ Q. -- Q._~ S.) (t,2) to(z) =T.-_~lO r~(z) = ( ~ . _ ~ - _ ~ : + (v._~ s . : r �9

In this paper we obtain some results on classifying the behavior of {Tn} in terms of the behavior of {t.). SCHWERDTFEGER [4] and TrlRON [5] have solved this probiem for the case where {T.} is a periodic sequence.

Let {T~} be generated by functional composition from a sequence {t~}. If one takes

bn+ z '

then a continued fraction can be considered as a sequence {T~ (0)} (see THRON [O]). This approach has been used extensively in the study of the analytic theory of continued fractions. Indeed, PIRANIAN and TI:Ro~, by introducing Theorem 1.t, hoped to derive tools that would be helpful in the s t u d y of continued fractions.

Sequences {T.} generated by sequences {t.} of Moebius transformations were also used at other times, for example by ScHuR [3] and by WEYI~ [7].

2. Results Concerning Case (a) of Theorem 1.1 In some of the subsequent theorems, it will be advantageous to normalize

the Tn in accord with the following theorem:

THEOREM 2.t. Let the sequence {T~} o/ Moebius trans/ormations converge tO a Moebius trans[ormation T. Let

To(~) - v .~ +.O. R n a + Sn '

Let T,, and T be normalized-as/ollows:

(2.t)

'(2.2)

Then

R z + S "

P n S n - - Q n R n = t , P S - - Q R = t .

I / P #O, la --~r/2<arg P.<z~/2, --.z~/2~arg P~z~]2.

I / P = O , let - ~r/2< argQ.<~r/2, - Jr/2< arg Q<-zr

lim P. = P , lim Qn = Q, lim Rn = R, lira S,, = S.

186 J . D . DE PREE and W. J. THRON:

PROOF. In proving the theorem, we consider the following cases: (i) R = 0, (ii) S = 0 , P : ~ 0 ; (iii) S = 0 , P = 0 ; (iv) P ~ 0 , R:~0, S :#0 ; (v) P = 0 , R ~ 0 , S ~ 0. Since the proofs of the various cases are similar, we will prove only case (iv).

From limT~(oo) = T(oo) and lim T~(0)= T(0) we obtain

(2,3) limP./R. = P/R, lim Q./S~ = Q/S.

Subtracting these we have l i m ( P . S . - - Q . R . ) / R . S ~ = ( P S . QR)/RS or

(2.4) l imR.S~=R S. Similarly

(2.5) IimP,~ Q,t= P Q.

Multiplying (2.3 a) and (2.4) one arrives at

(2.6) limPn S~ = P S. Similarly

(2.7) lim Q.,~R,, = QR.

From (2.3) and l imT.0 ) = T(t) we have limR,dQ,~=R/O and this with (2.5) implies

�9 (2.8) l i m P . R . = PR.

Multiplying (2.3 a) and (2.8) .we obtain limP.*----- P*. But from the restrictions on P. and P, we conclude that

(2.9) limP. = P .

It then follows from (2.5), (2.8), (2.6), and (2.9) that

lim O~ = Q, lim R.--- R, lim S~ = S.

THEOREM 2.2. Suppose the sequence {T.} o/Moebius trans/ormations converges to a Moebius trans/ormation T. Then the convergence is uniJorm on the Riemann sphere with respect to the chordal distance.

PROOF. The chordal distance e on the Riemann sphere is given by

IT. (z) - T(z) I Q (~ (z), T(z)) = V( t + IT.(01 ~) (t + I T0)I 2) '

(see [1, p. 811). The proof is by contradiction. If the convergence is not uniform, then there exists a sequence {Nk} of positive integers increasing monotonely to infinity and a sequence {zk} of complex numbers such that

O(TN,(Zk), T(zk) ) >=e o (k = t , 2, ...) for some e 0 > 0.

We distinguish two cases: (i) The sequence {zk} is bounded.

(ii) The sequence {zk} is unbounded.


(i) {z~} has a finite limit point ~. Replacing {z,} by a suitable subsequence we may assume zk-+$. We consider two subcases:

(a) [T(~)[ < oo, (b) T(~) = oo.

(a) Since 9 (T. (z), T ( z ) ) ~ IT. ( z ) - T(z)l, it will suffice to show that

[Tzc,(zk)- T(zk) [ _-->e 0, (k =1 , 2 . . . . )

leads to a contradiction. We have

ITN,(Zk)__T(zk)[= PN,~k+O2V~ Pzk+O RN, z~ + SNk R z~ + S

__ (.PN, R -- PRN) z~ + (PN, S -- q RN1 ' + (21vkR -- P SN1,)zk+ (ON, S -- q SN) (RN z ~ + SN) (Rz~ + S)

Since we may assume that the coefficients of T, and T have been normalized as in Theorem 2.1, it follows from that result that the limit of the above expression is zero, and this is a contradiction.

(b) If T(~)-----c~, the chordal distance 0 is given by

t

0(rN czk), = V, + IrN / k)l

In order to prove that the convergence is uniform, we wish to show that

lim Q (TNk (Zk), oo) ---- 0. k - + o o

Now T(~) ---- oo implies ~ = -- S/R, R :4: O. Then

- - ~ - 0 0

RNk zk + SNJRNk

by Theorem 2.t. The limit of the numerator is not zero because the limit transformation is non-singular. Consequently

lim ~ (TNk (Zk), co) = 0. /~ ---> o o

(ii) Suppose there is a subsequence {z,(~)} of {zk} such that zk(~)-->eo.

We again distinguish two sub-cases:

(a) I T(oo)l<o~; (b) T(oo)=oo.

The arguments are similar to case (i).

THEOREM 2. 3. Suppose the sequence {T,) is generated by /unctional composition [rom a sequence {t,} o/Moebius trans/ormations, suppose [urther that {T,} converges to a Moebius trans]ormation. Then lim t,(z)---z.

PROOF. Theorem 2.t and formula (t .2). Theorems 2.3, 2.4 and 3A represent an effort to characterize the behavior

of {T~) in terms of the behavior of {t,). It will be observed that Theorem 2.3 gives a necessary condition and Corollary 2.1 gives sufficient conditions on

188 J . D . DE PREE and W. J. THRON:

{t.) in order that {T.) satisfy case (a) of Theorem 1.1. Unfortunately necessary and sufficient conditions are not yet known. Theorem 3A gives sufficient conditions on {t.} - for {T.} to satisfy case (b) of Theorem tA. Necessary conditions remain unknown. We assume throughout the remainder of the paper that the relationships between the coefficients of T. and t. are given by the recursion formulas (t A).

THEOREM 2.4. Let P.z + Q.

r . ( z ) =t~o ... o t . ( z ) - - R . ~ + s . '

where

c, z + d, ' Then

(2.1o) ~,lP.-~,-~l, ~,IQ.-Q,-ll, ~,IR.-R.-ll, ~.d ~,IS.--S.-,I . = 2 . = 2 n=2 . = 2

,,zz con~er~e # and only i / H ~ . , ~, b., Z ~., ,~na Ha,, ~z1 converge absomely. PROOF. Suppose first that the series in (2.t0) all converge. Then the

sequences {P.), {Q.}, {R.), {S.) all converge. From the recursion formulas (1.1) we obtain

a .=P.S ._I - -R .Q._I - - - - t - - P . ( S . - - S._1) + R . ( Q . - - g.-1)

b. = Q. S._l - s . Q._I = s . ((2. - (2.-1) - (2. (s . - s,,_l)

c. = R . P . _ 1 -- P . R . _ I = P . (R. -- R._I) -- R . (]~ -- P.-1)

d . = P . _ I S . -- Q . R . _ I = I - - S . ( P . - P . _ I ) + (2 . (R. - -R. -1)

and it follows that H a . , F, b., F, c. and l i d . all converge absolutely.

Next suppose that H a . , Y, b., X c. and H d . are all absolutely convergent. Let a . = l + e. and d. = t + (~... Then it follows from the absolute convergence of H a . and I - ld . that ~][e.[ and ~,[0n[ converge. Using the recursion formulas (tA) and induction, we observe that ]P.], ]Q.], ]R.] and I S.] are bounded by

/"/0+ I~,1+ la, t + Ib, l + I~,1), �9 ~ ' = l

which co~verges, since Y. (1 ~1 + l a, l + I b, I + l e, I) converges. Again, from the recursion formulas we obtain

P . - P . - t = R - I * . + Q._~c. ,

Q,,- O , , - l=R- lb .+ Q._la.,

R. -- R . _ I = R . _ l e . + S._lc,,,

Sn -- S . _ I = R . _ l b . + S.•

and it follows that Z t P . - - P . _ I ] , Y, lq . -q . - lJ Y, IR.-R._ll Y, [ S. -- S,,-ll all converge.

and

On Sequences O f Moebius Transformat ions 189

CORALLARY 2.t. Let T.=tao ... o t . ,

w h e r e

t~ (z) - ~'* + b, a, d~ - - b , e~ 4= O. c~, z + dv '

I / H a , , Y, b,, Y. c, and H d , are all absolutdy convergent, then {T,} converges to a Moebius trans/ormation.

PROOF. From Cauchy's convergence criterion applied to Y. [P~-P~-xl and the inequality

n + p

[P,.+p--P~I < X [ P . + I - PP.[

it follows that {P.) converges. Similarly {Q.}. {R.) and {S.} also converge. L e t

lira P. = P , lira Q. = Q, lim R . - R, lira S. = S.

Then PS-:-QR=lim(P,S, Q,R,,)=limI-l(a,d,--b.c,)=A4:0. Thus {T,} converges to a Moebius transformation. .=1

3. Results concerning case (b) of Theorem 1.1

THEOREM 3.1. Let

where

T,(z),=tlo...ot,(z)- ~"~ + Q" R n z + S n '

t , (Z) - - a~, z + by a , d, - - b, c, : - i . c , z + d , '

T~ = T,,o 2 -1 o Tp*, where n = m + p -- 1, and

(3.1) ~* = t* o t* o . . . o t~, where t* = 2,

and t*=t,,+,_ t, 2 ~ v ~ p .

If it can be shown that lira T~(Z) = F * , a constant, for all z:4:0, it will then :b-+ oo

�9 follow by the continuity of Tmo2 -1, that lim Tn(x)=/ ' , a constant, f0r all z~=0. .-.o~

Suppose ]urther that

(i) ~ I b.I and ~. I b,.I converge. "t 1

0i) la.I = t + e., diverges. 1

Then lim T,. (z)=F, a constant,/or all z =4: O. ~--.> oo

PRooF. Let Tp*=2otm+lo ... otto.p_1,

where 2 is a Moebius transformation. I t is then clear that

190 J.D. DE PREE and W. J. THRON:

(3.2) and

(3.3) From (3.3) we have

(3 .-4)

From a.d,,--b,,c.---t, we obtain

Id.l_-<a Let

(3.5)

The theorem will be proved through the use of several lemmas, some o f whose proofs are by mathematical induction. Consequently, i t is desirable t o have a starting point for each of the inductive proofs. These starting points are obtained by suitable choices of the number m and the transformation 2.

From the convergence of Y, lb. I and Z lc, I foliows the existence of m > 0 such that

/ ~ ( t + 2 Ib, I) < ~ ~ = m + l

oo

v = r a + l

I~.1 ~. I~.1 < [b.I, n > m + t . ~=ra+l

1 + un

o~.=max{2 l b.l, *.}.

By considering the two cases en--[bnc.[ >0 (<0) and with the help of (3.4), we obtain

n--1

(3.6) Ib.l+lb.I Y, Ic . l+ la . l<t+~ . f ora l l~>~+ l . p=m+l

With this choice of m and with 2 (z)=z/(}z + t), we may assum e, without loss of generality, that we are dealing with sequences {T~} generated by sequences {t.} for wh.ich the following conditions hold: (i) and (ii) of Theorem 3-t,

(iii) s I c. I < ~" n = l

s - - 1

(iv) Ib.l+lb.I Z Ic.l+ld.I <a+, , . for all n. co

(v) HO+ 2Ib, I)<L

(vi) 1Pl1=t+~1<(t+~1)0+1c11), 1911--1b~1<t+~1,

IRll =1~1 __>(1+ e,)(t ~+~ 1~!), where a~, bt, Cl, da are obtained' from ;t (z), and ~x is given by (3.4).

With these conditions; the proof of the theorem is as follows:

On Sequences of Moebius Transformations 1 9 1

LEMMA 3.1. n

IO.I < / / 0+~ , )and ~ '~1

IS.I </'/(t+~,) aria Y = I

PROOF. From (vi) we have Assuming that

n - - 1

[ Qn-l[ N/ - / (1 + o:,) and

Ir and I~1=<(t+~)(t+1~1).

we obtain, using the recursion formulas t . t ,

IGI--- I b,P.~l + Id.Q.-~l Z(H(t+,,.) t+Zlc, b,l+l.d.lrt(t+,,.)=l

n

N H (t + e,), by (iv).

Also

___(t+~.)(H(t+~,) t+E1~,,=1 +[~.[H(t+=,),=~

z (:I(t + ~,)) (, +~! c,I). The proof of the second set of inequalities is similar.

LEMMA 3.2. The inequalities in Lemma 3.t imply

lim s . = 0 and lim Q" = 0 . n

" "'-'~ H O + ",) " - - ' ~ H 0 + ~-.)

PRooF. From the recursion formulas 0.1) we obtain

Sn=bnRn-l-]-dn. Sn-l= ~,, ( h dk)R,-lb~,q-hd.. ~ = 2 ~ k = ~ + l ]

Consequently "�9 ~,=1

n

H(t + a,) (3.6) , = 1

n n

/ / l a , I ~ 17 lakl < "; + ~-i

H ( ! + u.) ~__~ / - f (1 + ~k) H(I + ak) �9 =1 k=~ k=l

ld, l H 14[ = k = u + l < . + - ~ - - - - ~ [by[.

H 0 + q-,) ~=~ / / (1 + ~k) \k=x

192 J.D. DE PREE and W. J. THRON:

Since Id, I < t+[b,c~[ --<'t+ lb, c,I, where Y, Ib, c,I converges, and since t § ev - -

Y, 0~, = § 0% we have

Hldk l (3.7) lim ~=u = 0 .-.oo H ( I + ~)

k=v

for each ff and each v. Now let e > 0 be given. We then split the sum in (3.6) into two parts,

each of which is less than */3. This is possible because of (3.7) and the fact that Y. lb, I and Y, lc,] both converge, I t also follows from (3.7) that t h e first term on the right in (3.6) can be made less than e/3. Hence, there exists N > 0 such that n>=N implies

�9 I s.I < ./3 + ,/3 + ,/3 =*- [/(1 + ~,)

This establishes the first limit in Lemma 3.2. The proof of the second is similar.

�9 LEMM, 3"3.

PROOF. From condition ( v i ) w e have IRll>__(t§ Assuming the inequality to hold for n - t, we' ' ~ ' ' / o b t a i n t + e x

I R, I > ]Ia, R,_ll - - Ic , s , -xl l n--1 t-(/7 ' + ~ " " - ~ - - \ v : l 1 ~ - ~ / levi [CnIH({§

__ (t+~.) - ~ I .t = l c . \L-I ~Z-.] .=~/- /0+~')

l We see now that the removal of the absolute value bars is justified by conditions (iii) and (v) and the inequality

'+="- < / / ( a § Ib.t!. .=1 t § e. .=1

LEMMA 3.4. The sequence {P,,/R,} converges.

PROOI~. From the recursion formulas (t.t) we obtain

P./R,, - - P, , , x / R . _ x = -- c , , /R.R, ,_x, and from this

P.,/R,~-= P~/R~ - - X c , /R ,R ,_a .


From Lemma 3.3 we have that {R,) is bounded away from zero. convergence of ~. I c,[, it now follows that {Pst/R~} converges.

LEMMA 3" 5.- lim S./Rst = 0 and lim Qst/Rst = O.

PROOF. From the identity

and the definition of ~ , it follows that St st

/ - / ( t + ~ , ) ~ o H ( t + s~), where v = l v = l

Using Lemma 3.3, we obtain

o = t + ~+s . / St='l

n

1~ (1 4- o~v) co ~I (1 4- ~I,) I S . I R . I _ I s. I ,= , < I s. I ~=,

HO+~,) IR"I / /0+~,1 IR"I

[ Sn[ co St �9 OO

/ / ( t + ~,,) t - ~Y, lc,,I v=1 v = l

=

Since 1--o~. [c~[ >0, it follows from Lemma3.2 that l i m S s t / R , = O . proof of the second limit is similar.

Finally, write T . (z) = (P./R.) z + (Q./R.)

z + (S. /R.)

Then using Lemmas 3.4 and 3-5, we obtain

l i m T . (z) = F for a l l z # 0,

where F - - limPdR...This'c~ the proof of Theorem 3.1.

From the

The

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1954. [ 2 ] P I R A N I A N , G., and W: J. THRON: Convergence Properties of Sequences of Linear

Fract ional Transformations. Mich. Math. J. 4, 1 2 9 - t 35 (t957). [8] ScouR, J. : Ober Potenzreihen, die im Innern des Einheitskreises beschr&nkt sind.

J. Reine Angew. Math. 147, 205--232 (19t7), [4] SCHWERDTFEGER, H. : Moebius Transformations and Continued FractiOns. Bull. Amer.

Math. Soc. 52, 307--309 (t946). [5] THROW, W. J . : Lecture Notes from Seminar in Continued Fractions. (Unpublished,

1956). [6] -- Convergence Regions for Continued Fractions and Other Infini te Processes. Amer.

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Dept. o/Math., Oregon State University, Corvallis, Oregon (U. S.A .) Dept. of Math., University o] Colorado, Boulder, Colorado (U.S.A )

(Received July 6, 1962) M a t h e m a t i s c h e Ze i t sch r i f t . Bd . 80 t 4

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On sequences of Moebius transformations